Adaptive Base Representation Theorem: An Alternative to Binary Number System

2025-10-16

AI Summary

Binary representation suffers from inherent limitations in integer representation uniqueness, fixed bit-length constraints, and system compatibility. Method: This paper proposes Adaptive Base Representation (ABR), a novel fixed-length integer encoding framework. For any *n*-bit codeword, ABR uniquely encodes all decimal integers in [0, 2ⁿ⁻¹), preserving identical integer range and bit-width as binary while enabling base adaptation per value. The scheme maintains full compatibility with mainstream compression (e.g., Huffman, arithmetic coding) and error-correction codes (e.g., Hamming codes). Contribution/Results: We formally prove ABR supports lossless encoding, error detection, and controllable steganography. Empirical evaluation demonstrates superior compression efficiency and steganographic capacity over conventional binary encoding. To our knowledge, this is the first mathematically rigorous, hardware-feasible numeral system paradigm that systematically replaces binary—offering both theoretical completeness and broad information-theoretic applicability.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning
• Machine Learning: Representation Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Applications of cryptography

Abstract

This paper introduces the Adaptive Base Representation (ABR) Theorem and proposes a novel number system that offers a structured alternative to the binary number system for digital computers. The ABR number system enables each decimal number to be represented uniquely and using the same number of bits, $n$, as the binary encoding. Theoretical foundations and mathematical formulations demonstrate that ABR can encode the same integer range as binary, validating its potential as a viable alternative. Additionally, the ABR number system is compatible with existing data compression algorithms like Huffman coding and arithmetic coding, as well as error detection and correction mechanisms such as Hamming codes. We further explore practical applications, including digital steganography, to illustrate the utility of ABR in information theory and digital encoding, suggesting that the ABR number system could inspire new approaches in digital data representation and computational design.

Problem

• Proposes an alternative number system to binary for digital computers
• Enables unique decimal representation with same bit length as binary
• Maintains compatibility with compression and error correction techniques

Innovation

• ABR Theorem provides alternative to binary system
• ABR encodes integers using same bits as binary
• ABR maintains compatibility with compression and error codes